On the Lagrangian structure of quantum fluid models

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April 2014, 34(4): 1375-1396. doi: 10.3934/dcds.2014.34.1375

On the Lagrangian structure of quantum fluid models

Philipp Fuchs ^1, , Ansgar Jüngel ^1, and Max von Renesse ^2,

1.	Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Wien, Austria, Austria
2.	Institute for Mathematics, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

Received January 2013 Revised May 2013 Published October 2013

Abstract
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Some quantum fluid models are written as the Lagrangian flow of mass distributions and their geometric properties are explored. The first model includes magnetic effects and leads, via the Madelung transform, to the electromagnetic Schrödinger equation in the Madelung representation. It is shown that the Madelung transform is a symplectic map between Hamiltonian systems. The second model is obtained from the Euler-Lagrange equations with friction induced from a quadratic dissipative potential. This model corresponds to the quantum Navier-Stokes equations with density-dependent viscosity. The fact that this model possesses two different energy-dissipation identities is explained by the definition of the Noether currents.

Keywords: magnetic Schrödinger equation, symplectic form, Noether theory, quantum Navier-Stokes equations, osmotic velocity., Geometric mechanics.

Mathematics Subject Classification: Primary: 35Q40, 37J10, 58D3.

Citation: Philipp Fuchs, Ansgar Jüngel, Max von Renesse. On the Lagrangian structure of quantum fluid models. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1375-1396. doi: 10.3934/dcds.2014.34.1375

References:

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2008.
[2]	A. Arnold, Mathematical properties of quantum evolution equations, in "Quantum Transport Modelling, Analysis and Asymptotics'' (eds. G. Allaire, A. Arnold, P. Degond and T. Y. Hou), Lecture Notes in Mathematics, 1946, Springer, Berlin, (2008), 45-109. doi: 10.1007/978-3-540-79574-2_2.
[3]	V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.
[4]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.
[5]	S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297.
[6]	B. van Brunt, "The Calculus of Variations,'' Universitext, Springer-Verlag, New York, 2004.
[7]	P. Dirac, The Lagrangian in quantum mechanics, Phys. Z. Sowjet., 3 (1933), 64-72.
[8]	Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics, Acta Mech., 23 (1975), 17-27.
[9]	E. Feireisl, "Dynamics of Viscous Compressible Fluids,'' Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.
[10]	J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318-390. doi: 10.1016/j.matpur.2011.11.004.
[11]	L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812567635.
[12]	T. Frankel, "The Geometry of Physics. An Introduction,'' Cambridge University Press, Cambridge, 1997.
[13]	G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control, Intern. J. Tomogr. Stat., 5 (2007), 109-114.
[14]	J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems, Phys. Lett. A, 317 (2003), 255-259. doi: 10.1016/j.physleta.2003.08.028.
[15]	A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.
[16]	A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.
[17]	A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, J. Stat. Phys., 145 (2011), 1661-1673. doi: 10.1007/s10955-011-0388-3.
[18]	A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution, Kinetic Related Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785.
[19]	J. Lafferty, The density manifold and configuration space quantization, Trans. Amer. Math. Soc., 305 (1988), 699-741. doi: 10.1090/S0002-9947-1988-0924776-9.
[20]	J. Lott, Some geometric calculations on Wasserstein space, Commun. Math. Phys., 277 (2008), 423-437. doi: 10.1007/s00220-007-0367-3.
[21]	E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1926), 322-326. doi: 10.1007/BF01400372.
[22]	P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit, J. Funct. Anal., 259 (2010), 1542-1576. doi: 10.1016/j.jfa.2010.05.013.
[23]	R. McCann, Polar factorization of maps on Riemannian manifolds, GAFA Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.
[24]	E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.
[25]	F. Otto, The geometry of dissipative evolution equations: The porous-medium equation, Commun. Part. Diff. Eqs., 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[26]	F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.
[27]	M.-K. von Renesse, On optimal transport view on Schrödinger's equation, Canad. Math. Bull., 55 (2012), 858-869. doi: 10.4153/CMB-2011-121-9.
[28]	W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics, SIAM Review, 23 (1981), 467-494. doi: 10.1137/1023098.
[29]	R. Talman, "Geometric Mechanics,'' Wiley, New York, 2000.

show all references

References:

[1]	L. Ambrosio, N. Gigli and G. Savaré, "Gradient Flows in Metric Spaces and in the Space of Probability Measures,'' Second edition, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2008.
[2]	A. Arnold, Mathematical properties of quantum evolution equations, in "Quantum Transport Modelling, Analysis and Asymptotics'' (eds. G. Allaire, A. Arnold, P. Degond and T. Y. Hou), Lecture Notes in Mathematics, 1946, Springer, Berlin, (2008), 45-109. doi: 10.1007/978-3-540-79574-2_2.
[3]	V. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), 319-361. doi: 10.5802/aif.233.
[4]	J.-D. Benamou and Y. Brenier, A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem, Numer. Math., 84 (2000), 375-393. doi: 10.1007/s002110050002.
[5]	S. Brull and F. Méhats, Derivation of viscous correction terms for the isothermal quantum Euler model, Z. Angew. Math. Mech., 90 (2010), 219-230. doi: 10.1002/zamm.200900297.
[6]	B. van Brunt, "The Calculus of Variations,'' Universitext, Springer-Verlag, New York, 2004.
[7]	P. Dirac, The Lagrangian in quantum mechanics, Phys. Z. Sowjet., 3 (1933), 64-72.
[8]	Dj. Djukic and B. Vujanović, Noether's theory in classical nonconservative mechanics, Acta Mech., 23 (1975), 17-27.
[9]	E. Feireisl, "Dynamics of Viscous Compressible Fluids,'' Oxford Lecture Series in Mathematics and its Applications, 26, Oxford University Press, Oxford, 2004.
[10]	J. Feng and T. Nguyen, Hamilton-Jacobi equations in space of measures associated with a system of conservation laws, J. Math. Pures Appl. (9), 97 (2012), 318-390. doi: 10.1016/j.matpur.2011.11.004.
[11]	L. Brown, ed., "Feynman's Thesis. A New Approach to Quantum Theory,'' World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2005. doi: 10.1142/9789812567635.
[12]	T. Frankel, "The Geometry of Physics. An Introduction,'' Cambridge University Press, Cambridge, 1997.
[13]	G. Frederico and D. Torres, Nonconservative Noether's theorem in optimal control, Intern. J. Tomogr. Stat., 5 (2007), 109-114.
[14]	J.-L. Fu and L.-Q. Chen, Non-Noether symmetries and conserved quantities of nonconservative dynamical systems, Phys. Lett. A, 317 (2003), 255-259. doi: 10.1016/j.physleta.2003.08.028.
[15]	A. Jüngel, "Transport Equations for Semiconductors,'' Lecture Notes in Physics, 773, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-540-89526-8.
[16]	A. Jüngel, Global weak solutions to compressible Navier-Stokes equations for quantum fluids, SIAM J. Math. Anal., 42 (2010), 1025-1045. doi: 10.1137/090776068.
[17]	A. Jüngel, J. L. López and J. Montejo-Gámez, A new derivation of the quantum Navier-Stokes equations in the Wigner-Fokker-Planck approach, J. Stat. Phys., 145 (2011), 1661-1673. doi: 10.1007/s10955-011-0388-3.
[18]	A. Jüngel and J.-P. Milišić, Full compressible Navier-Stokes equations for quantum fluids: Derivation and numerical solution, Kinetic Related Models, 4 (2011), 785-807. doi: 10.3934/krm.2011.4.785.
[19]	J. Lafferty, The density manifold and configuration space quantization, Trans. Amer. Math. Soc., 305 (1988), 699-741. doi: 10.1090/S0002-9947-1988-0924776-9.
[20]	J. Lott, Some geometric calculations on Wasserstein space, Commun. Math. Phys., 277 (2008), 423-437. doi: 10.1007/s00220-007-0367-3.
[21]	E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Phys., 40 (1926), 322-326. doi: 10.1007/BF01400372.
[22]	P. Markowich, T. Paul and C. Sparber, Bohmian measures and their classical limit, J. Funct. Anal., 259 (2010), 1542-1576. doi: 10.1016/j.jfa.2010.05.013.
[23]	R. McCann, Polar factorization of maps on Riemannian manifolds, GAFA Geom. Funct. Anal., 11 (2001), 589-608. doi: 10.1007/PL00001679.
[24]	E. Nelson, Derivation of the Schrödinger equation from Newtonian mechanics, Phys. Rev., 150 (1966), 1079-1085. doi: 10.1103/PhysRev.150.1079.
[25]	F. Otto, The geometry of dissipative evolution equations: The porous-medium equation, Commun. Part. Diff. Eqs., 26 (2001), 101-174. doi: 10.1081/PDE-100002243.
[26]	F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361-400. doi: 10.1006/jfan.1999.3557.
[27]	M.-K. von Renesse, On optimal transport view on Schrödinger's equation, Canad. Math. Bull., 55 (2012), 858-869. doi: 10.4153/CMB-2011-121-9.
[28]	W. Sarlett and F. Cantrijn, Generalization of Noether's Theorem in classical mechanics, SIAM Review, 23 (1981), 467-494. doi: 10.1137/1023098.
[29]	R. Talman, "Geometric Mechanics,'' Wiley, New York, 2000.

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